# Differential Equation question

• Feb 10th 2008, 06:53 AM
keith
Differential Equation question
The problems in the section of the book list the questions in this way:
dy/dx = 4y+e^4xsin(5x) and y(0)=1
First I get P(x)= -4 and then Q(x) = e^4x sin(5x)
then we put it in standard form and put the y on the left and the x on the right. Take the derivative first and then integrate. I understand the problems from #'s 18 on but the first 18 are set up like this:
y'-2xy= e^x^2 In the directions it states: primes denote derivatives with respect to x. The teacher explained the part in the book and with his example I understood and could do the later problems but I guess sometimes the hang up is what the question is asking. If someone has time to show me an example of the type y' -2xy= e^x^2 I would appreciate it. I assume we would do the same, get y on the left and x on the right and follow the same steps but not sure how to start. It seems that math is seeing patterns and if I could see one problem worked out I should be able to see the pattern and finish the rest of them.
Thank You,
Keith
• Feb 10th 2008, 07:09 AM
Krizalid
Well, we're talking about linear ODEs. Do you know what the integrating factor is?

Quote:

Originally Posted by keith
The problems in the section of the book list the questions in this way:
dy/dx = 4y+e^4xsin(5x) and y(0)=1

This is $\displaystyle y'-4y=e^{4x}\sin5x.$ (See my signature for LaTeX typesetting.)

Your integrating factor is $\displaystyle e^{-4x},$ so just multiply the entire equation by this term and go from there.

Quote:

Originally Posted by keith
If someone has time to show me an example of the type y' -2xy= e^x^2 I would appreciate it.

The same idea: your integrating factor in this case is given by $\displaystyle \mu(x)=\exp\left\{\int-2x\,dx\right\}.$ Get it and do similar things with the first one.